Directory UMM :Journals:Journal_of_mathematics:TAC:
Theory and Applications of Categories, Vol. 2, No. 7, 1996, pp. 69{89.
COMBINATORICS OF CURVATURE, AND THE BIANCHI
IDENTITY
ANDERS KOCK
Transmitted by F. William Lawvere
ABSTRACT. We analyze the Bianchi Identity as an instance of a basic fact of combinatorial groupoid theory, related to the Homotopy Addition Lemma. Here it becomes
formulated in terms of 2-forms with values in the gauge group bundle of a groupoid,
and leads in particular to the (Chern-Weil) construction of characteristic classes. The
method is that of synthetic di erential geometry, using \the rst neighbourhood of the
diagonal" of a manifold as its basic combinatorial structure. We introduce as a tool a
new and simple description of wedge (= exterior) products of di erential forms in this
context.
Introduction
We shall give a proof of the Bianchi Identity in di erential geometry. This is an old
identity the novelty of the proof we present is that it derives the identity from a basic
identity in combinatorial groupoid theory, and permits rigorous pictures to be drawn of
the mathematical objects (connections, curvature,...) that enter.
The method making this possible here, is that of synthetic di erential geometry (\SDG" see e.g. Koc81]), which has been around for more than 20 years but in contrast
to most of the published work which utilizes this method, the basic notion in the present
paper is the rst order notion of neighbour elements x y in a manifold M , rather than
the notion of tangent vector D ! M (which from the viewpoint of logic is a second order
notion). This latter notion really belongs to the richer world of kinematics, rather than
to geometry: tangent vectors are really motions.
In so far as the the specic theory of connections is concerned, it has been dealt with
synthetically both from the viewpoint of neighbours, and from the viewpoint of tangents
for the latter, cf. KR79], MR91], and more recently and completely, Lav96], (which also
contains a proof of the Bianchi identity, but in a completely di erent spirit).
On the other hand, the theory of connections, and the closely related theory of differential forms, has only been expounded from the combinatorial/geometric (neighbour-)
viewpoint in early unpublished work by Joyal, and by myself, Koc80], Koc83], Koc82],
Koc81] (section I.18), Koc85] but it has only been mentioned in passing in the monographs MR91] (p. 384), Lav96] (p. 139 and 180).
Received by the editors 20 June 1996 and, in revised form, 6 September 1996.
Published on 26 September 1996
1991 Mathematics Subject Classication : 58A03, 53C05, 18F15 .
Key words and phrases: Connection, curvature, groupoid, rst neighbourhood of the diagonal.
c Anders Kock 1996. Permission to copy for private use granted.
69
Theory and Applications of Categories,
Vol. 2, No. 7
70
My contention is that the combinatorial/geometric viewpoint comes closer to the geometry of some situations, and permits more pictures to be drawn. One may compare
the theory of di erential forms, as expounded here, (and in Koc82] and Koc81] Section
I.18.)), with the theory of di erential forms as seen from the viewpoint of tangents, and
expounded synthetically in KRV80], Koc81] Section I.14, MR91], Min88], Lav96].
Thus, the fact that di erential k-forms (k 2) should be alternating is something that
is deduced in the combinatorial/geometric approach, but is postulated in the more standard
synthetic approach (as in Koc81], Def. I.14.2). Likewise, the denition of wedge product
of forms as presented in the present paper from the combinatorial/geometric viewpoint
(for the rst time, it seems), is more self-explanatory than the theory of Min88] and
Lav96] p. 123 (their theory is here anyway quite close to the \classical" formalism).
The piece of drawable geometry, which, in the approach here is the main fact behind
the Bianchi identity, is the following: consider a tetrahedron. Then the four triangular
boundary loops compose (taken in a suitable order) to a loop which is null-homotopic inside
the 1-skeleton of the tetrahedron (the loop opposite the rst vertex should be \conjugated"
back to the rst vertex by an edge, in order to be composable with the other three loops).
- This fact is stated and proved in Theorem 9.1 (in the context of combinatorial groupoid
theory). Ronald Brown has pointed out to me that this is the Homotopy Addition Lemma,
in one of its forms, cf. e.g. BH82] (notably their Proposition 2).
In so far as the di erential-geometric substance of the paper is is concerned (i.e.
leaving the question of method and pictures aside), the content of the paper is part of the
classical theory of connections in vector bundles, leading to the Chern-Weil construction
of characteristic classes for vector bundles. I learned this material from Mad88] Section
11, (except for the groupoids, which in this context are from Ehresmann) cf. also MS74].
One cannot say that I chose to couch this theory into synthetic terms this is not a matter
of choice, but rather a necessity to see what connection theory means in terms of the
neighbour relation, now that this relation has forced itself onto our minds.
I would like to acknowledge some correspondence with Professor van Est in 1991, where
he suggested a relationship between combinatorial groupoid theory and the neighbourhood
combinatorics of synthetic di erential geometry. He also sent me his unpublished vE76],
where in particular some di erential-geometric terminology (curvature, Bianchi identity,..)
is used for notions in combinatorial groupoid theory.
1. Preliminaries
We shall need to recall a few notions from synthetic di erential geometry. The main thing
is that for any manifold M , there is, for each integer k, a notion of when two elements
x y 2 M are k-neighbours, denoted x k y. In particular, x 1 y is denoted x y. (The
\set" of pairs (x y) 2 M M with x y is what in algebraic geometry is called the rst
neighbourhood of the diagonal, or M(1) if M is an ane scheme Spec(A), then M M
is Spec(A A) and M(1) = Spec((A A)=I 2) where I is the kernel of the multiplication
map A A ! A.) There exist categories containing the category of smooth manifolds as
Theory and Applications of Categories,
Vol. 2, No. 7
71
a full subcategory, but in which any manifold acquires such relations like and k
as new subobjects of one talks about these new (sub-)objects in terms of their
\elements", although (1), say, does not have any more points than itself does.
The set of -neighbours of an element 2 is denoted Mk ( ) and called the -monad
around . The union of all the -monads around is denoted M1( ).
In particular, the -monad around 0 2 Rn , i.e. the \set" (object) of -neighbours of
0 2 Rn is denoted k ( ) if = 1, we just write ( ). So
M
M
M
M
M
k
x
x
M
k
x
x
k
x
k
D
k
n
k
D n
M1(0) = ( )
D n :
( ) Rn as the set
One may also describe
Dk n
f( 1 . . .
d
dn
) j the product of any + 1 of the i's is 0g
k
d
:
Let us explicitly note the following property of ( ): If any multilinear function has
an element from ( ) as argument in two di erent places, then the value is zero.
The sub\set" k ( ) Rn is one of the important kinds of \innitesimal" objects in
SDG. Another one is k ( ) Rn . . . Rn ( times) which we shall describe now.
First: an innitesimal -simplex in a manifold is a +1-tuple of elements ( 0 . . . k ),
such that for any , i j . Such a simplex is called degenerate if two of the vertices
i and j are equal ( 6= ).
A -tuple of elements ( 1 . . . k ) in Rn is said to belong to k ( ) if the + 1-tuple
(0 1 . . . k ) is an innitesimal -simplex so not only are all i in ( ), but also each
n
n
i ; j . This innitesimal subobject of R . . . R ( times) was introduced in Koc81]
there it was denoted ~ ( ).
From Koc81] Section I.18, we quote the following result, which seems not to have
been considered in any of the subsequent treatises on SDG, but which is crucial to the
following.
1.1. Theorem. Any map : k ( ) ! R which for each = 1 . . . has the property
that ( 1 . . . k ) = 0 if i is 0, is the restriction of a unique -linear alternating map
: (Rn )k ! R.
From the Theorem, we may immediately deduce a more general one, where the
codomain R is replaced by any other nite dimensional vector space .
D n
D n
D
n
D n
k
k
i j
x
x
i
k
d
d
M
x
k
x
x
x
j
d
d
d
D n
k
d
d
k
D n
k
D k n
!
! d
d
D n
i
d
k
k
!
V
2. Logarithms on General Linear Groups
Let
be a nite dimensional vector space (of dimension , say). Then we have the ring
( ), whose additive group is a vector space of dimension 2, and whose group of units
is a Lie group, denoted ( )
( ). The identity element 2 ( ) is denoted .
If k 0 in
( ), it follows by matrix calculations (identifying with Rq , and
( ) with the vector
space
of matrices) that k+1 is the zero endomorphism, so
P
1
ap
the exponential series 0 p! has only nitely many non-zero terms, and it is actually an
automorphism ! (due to the rst term which is ) it is thus an element of ( ),
V
q
E nd V
q
GL V
a
E nd V
GL V
E nd V
E nd V
V
q
V
V
e
q
a
e
GL V
Theory and Applications of Categories,
Vol. 2, No. 7
72
and it is denoted exp(a). Similarly, if b k e2 in GL(V3 ), log(b) 2 End(V ) is dened by
the logarithmic series log(b) = (b ; e) ; (b;2e) + (b;3e) ; . . ., which likewise terminates.
The fact that exp and log establish mutually inverse bijections Mk (0) = Mk (e) for all k,
and hence M1(0) = M1(e), is the standard power series calculation. Likewise, if a and
b are commuting endomorphisms in M1 (0) End(V ), the formula
exp(a + b) = exp(a) exp(b)
(1)
comes by a standard calculation with series. Similarly for log on commuting elements in
End(V ).
In particular, if V is 1-dimensional, we get the standard homomorphisms exp : (R +) !
(R ) and log (natural logarithm), which here happen to be extendible beyond the respective 1-monads, and also are group homomorphisms.
The following is a basic fact in Lie theory:
2.1. Proposition. Let and be e 2 GL(V ). Then the logarithm of the group
theoretic commutator is the ring theoretic commutator of their logarithms.
Proof. Let a and b be the logarithms of and , respectively, so = e + a. From
e
follows a 0 2 End(V ). Similarly for b. Using coordinate calculations in the algebra of
q q matrices, it is immediate from a 0 that a a = 0, and hence that the multiplicative
inverse of = e + a is e ; a similarly for . Thus
;1 ;1 = (e + a)(e + b)(e ; a)(e ; b)
which multiplies out in End(V ) by distributivity to give 16 terms. Some of these, like aba
contain a repeated a or b factor, and are therefore 0, again by matrix calculations, using
a 0, resp. b 0. The \rst order" tems are a b ;a and ;b, so they cancel. Also aa
and bb vanish, and we are left with e + ab ; ba since ab ; ba 0 (being 0 if a = 0), we
get that log(;1 ;1) = ab ; ba = a b], proving the Proposition.
In particular, 1-neighbours of e in GL(V ) commute if and only if their logarithms do
in End(V ) (this actually also holds for k-neighbours of e). Among pairs which always
commute in End(V ) are pairs of form (d) (d), where and are linear maps Rm !
End(V ) and d 2 D(m). This again follows from matrix calculations.
We shall give a sample of other commutation laws that can be derived from this
principle, and which are to be used later. Assume f and g are bilinear maps Rm Rm !
End(V ), and assume 0 d1 d2 d3 form an innitesimal 3-simplex in Rm. Then f (d1 d2)
commutes with g(d1 d3) as well as with g(d2 ; d1 d3 ; d1). For the former, this follows
by keeping d2 and d3 xed then each of the two elements depend linearily on d1, and
we are back at the previous situation similarly, for the second assertion, we rewrite
g(d2 ; d1 d3 ; d1) as a sum of four terms, using bilinearity of g then each of the four
terms has either a d1 or a d2 factor, and hence commutes with f (d1 d2).
The results of this section generalize to \abstract" Lie groups G, without representing
these as subgroups of some GL(V ) this depends on consideration of fragments of the
Theory and Applications of Categories,
Vol. 2, No. 7
73
Campbell-Baker-Hausdor series. In particular, if exp : D(p) ! M1(e) G is any
bijection taking 0 to e, additive inversion v 7! ;v in D(p) corresponds to multiplicative
inversion in G.
3. Dierential forms with values in Lie groups
Let M be a manifold, and let G be a Lie group. For k = 0 1 . . ., we dene a dierential
k-form with values in G to be a law ! which to each innitesimal k-simplex in M
associates an element !() 2 G the only axiom is that if the simplex is degenerate,
then !() = e, the neutral element of G. (Remark: it even suces to assume that
!() = e for any which is degenerate in the special way that it contains the vertex x0
repeated.)
3.1. Proposition. Let ! be a dierential k-form on a manifold M with values in a Lie
group G. Then ! is alternating, in the sense that swapping two of the vertices in the
simplex implies a \sign change":
!(x0 . . . xi . . . xj . . . xk ) = !(x0 . . . xj . . . xi . . . xk );1
(2)
for i 6= j .
m
Proof. Since the question is local, we may assume that M = R . Also, since values
of di erential forms are always 1-neighbours of the neutral element in the value group
G, only M1(e) G is involved and M1(G) = M1(0) Rn , i.e. M1(e) = D(n),
where n is the dimension of G. Now, (as mentioned above), under this isomorphism, the
inversion g 7! g;1 in M1(e) corresponds to the inversion v 7! ;v in D(n). Therefore,
the problem reduces to the case where M = Rm and G = Rn (under addition). Now any
innitesimal k-simplex (x0 . . . xk ) in Rm may be written (x0 x0 + d1 . . . x0 + dk ), where
(d1 . . . dk ) 2 k D(m). Write
!(x0 x0 + d1 . . . x0 + dk ) = f (x0 d1 . . . dk ):
(3)
Then for each x0, f (x0 ; . . . ;) is a function k (D(m)) ! Rn, (with value zero if one
of the blanks is lled with a zero), and hence, by Theorem 1.1, it extends to a k-linear
alternating function (Rm )k ! Rn, which we also denote f (x0 ; . . . ;). From this, it
is clear that we get a minus on the value of !(x0 x1 . . . xk ) if xi is swapped with xj
for i j 1, since this amounts to swapping di and dj . (For the same reason, it is also
clear that we get 0 if xi = xj for some i 6= j from this, the parenthetical remark prior
to the statement of the Proposition follows.) The argument for the case of swapping x0
with one of the other xi's is a little di erent. Without loss of generality, we may consider
swapping x0 and x1 = x0 + d1. This we do by a Taylor expansion in the rst variable for
the function f : Rm (Rm)k ! Rn , which is multilinear in the last k arguments. The
values to be compared are f (x0 d1 d2 . . . dk ) and f (x0 + d1 ;d1 d2 ; d1 . . . dk ; d1),
and the Taylor development of this latter expression gives
f (x0 ;d1 d2 ; d1 . . . dk ; d1) + D1f (x0 ;d1 d2 ; d1 . . . dk ; d1)(d1)
Theory and Applications of Categories,
Vol. 2, No. 7
74
where D1f denotes the di erential of f in the rst variable it is linear in the original k
variables as well as in the extra variable. Because d1 appears in two places in the k + 1
linear expression D1f , this term vanishes, so we are left with f (x0 ;d1 d2 ;d1 . . . d ;d1)
which we expand into a sum, using its multilinearity. In this sum, all but one term
contain a ;d1 as argument in two di erent places, so vanishes. The remaining term is
f (x0 ;d1 . . . d ) = ;f (x0 d1 . . . d ) = ;!(x0 x1 . . . x ), as desired. This proves the
Proposition.
Remark. It is tempting, (but unjustied without reference to Theorem 1.1), to attempt
to start the argumentation: \keep x2 . . . x xed, and consider !(; ; x2 . . . x ) as a
function of the two variables x0 and x1 = x0 + d1 only as a function f of d1, it extends by
the basic axiom of SDG to a linear map on the whole vector space". This is unjustied
for, it would require that f is already dened on the whole of D(m), but d1 is not free to
range over the whole of D(m), since it is still tied by the condition that x0 + d1 should
be neighbour to all the x2 . . . that we are keeping xed.
The fact that the \alternating" property comes so easily, namely just by proving that
the value on degenerate simplices is trivial, leads to a very simple description of wedge
product of forms this we deal with in Section 5.
There is an evident way to multiply together two k-forms on M with values in the
group G:
(! )(x0 . . . x ) := !(x0 . . . x ) (x0 . . . x )
The set of k-forms in fact becomes a group (M G) under this multiplication the neutral
element is the \zero form" z given by z(x0 . . . x ) = e.
k
k
k
k
k
k
k
k
k
k
k
4. Coboundary (commutative case)
For an innitesimal k +1-simplex = (x0 . . . x +1) in the manifold M , its i'th face @ ()
(i = 0 . . . k + 1) is the innitesimal k-simplex obtained by deleting the vertex x . If now
! is a k-form with values in the vector space V , we dene its coboundary d! to be the
k + 1 form given by the usual simplicial formula
k
i
i
d!() =
X(;1) !(@ ()):
k +1
i
i=0
(4)
i
The fact that this expression vanishes when two vertices in are equal is clear: all but
two of the terms in the sum contain two equal vertices, so vanish since ! is a di erential
form, and the remaining two terms cancel out because of the sign (;1) and because ! is
alternating, by Proposition 3.1.
Clearly, d is a linear map (M V ) ! +1 (M V ).
The following fact is the standard calculation from simplicial theory, so the proof will
not be given.
i
k
k
Theory and Applications of Categories,
Vol. 2, No. 7
75
4.1. Proposition. For any k-form !, d(d(!)) = 0, the zero k + 2 form.
So we have the deRham complex (M V ) for M (with coecients in the vector space
V ). But formally, it is more like an Alexander-Spanier cochain complex from topology.
5. Wedge products
Let ! be a k-form on a manifold M with values in (the additive group of) a vector space
U , and let be an l-form on M with values in a vector space V . If now U V ! W is
a bilinear map into a third vector space W , we can manifacture a k + l-form ! ^ on M
with values in W as follows. We put
(! ^ )(x0 . . . xk+l ) := !(x0 . . . xk) (x0 xk+1 . . . xk+l ):
(5)
To see that it is a form, we have to see that the value is 0 if two of the x's are equal. Since
the question is local, it suces to consider the case where M = Rm and x0 = 0. When ! is
considered as a function of the remaining (x1 . . . xk ) 2 k D(m), it extends by Theorem
1.1 to a multilinear map ! : (Rm)k ! V , and similarly extends to a multilinear , and
then since is bilinear, the whole expression in (5) is multilinear in the the arguments
x1 . . . xk+l, thus vanishes if two of them are equal (the xi's being 1-neighbours of 0).
Remark. The reason we have chosen to let the -factor in (5) have an x0 in its rst
position, rather than the aesthetically more pleasing xk , is that then the formula also
makes sense for forms with values in vector bundles U , V and W over M , with now
being a brewise bilinear map. Then both factors in (5) are in the same bre, namely the
one over x0. It can be proved that for forms with values in constant bundles, we get the
same value in either case. For ease of future references, we display also the aesthetically
pleasing denition (equivalent to the above one, for constant bundles)
(! ^ )(x0 . . . xk+l ) := !(x0 . . . xk ) (xk xk+1 . . . xk+l):
(6)
It should also be remarked that our wedge product agrees with the \small" classical
one: there are two conventions on how many k!, l! and (k + l)! to apply. The \big" one
gives the determinant as dx1 ^ . . . ^ dxp, the small one gives p1! times the determinant as
dx1 ^ . . . ^ dxp, in other words, the volume of a simplex, rather than the volume of the
parallellepiped it spans. With the emphasis on simplices in our treatment, this is anyway
quite natural.
Since our notions (form, coboundary, wedge) agree with the classical ones in contexts
where the comparison can be made (models of SDG), the calculus of di erential forms
(d being an antiderivation w.r.to ^ etc.) holds, so there is no need to prove them in the
context here, except for simplication this task I therefore postpone - I believe it will be
calculations that one could copy from those of Alexander-Spanier cohomology. (Also, our
notions agree with the \linearized" ones employed in other treatises of forms in SDG, say
Koc81], MR91], Min88], or Lav96] this is essentially argued in Koc81], Section I.18,,
except for the wedge.)
Theory and Applications of Categories,
Vol. 2, No. 7
76
6. Fibre bundles
Let be a groupoid over (i.e. with for its space of objects). To make a space over ,
! , into a bre bundle for , means to0 provide it with a left action by . Similarly
it makes sense to talk about a map ! being a bre bundle homomorphism w.r. to
bre bundle structures for . If ! is a group bundle, there is an evident notion
that the -action consists of group homomorphisms, so is a \bre bundle of groups", or
a \group bundle" for . Similarly for bundles of vector spaces, etc., so that we have a
notion of \bre bundle of vector spaces", or just vector bundle, for .
To the groupoid is associated the group bundle consisting of its vertex groups ( )
for 2 it is actually a bre bundle of groups for , because arrows of acts from
the left on the endo-arrows of by conjugation: if 2 ( ) and : ! is an arrow
in , then f :=
;1 2 ( ). This group bundle for is what in the literature
Mac89] is called the gauge group bundle of we denote it
( ).
If ! ! is a homomorphism of groupoids over , any bre bundle for ! canonically
can be viewed as a bre bundle for also (in analogy with \restriction of scalars" for
modules over rings). In particular, the gauge group bundle of ! is canonically a group
bre bundle for .
M
E
M
M
M
E
E
E
M
x x
x
M
f
f
x x
f
x
y
y y
gauge
M
7. Dierential forms with bundle values
Let be a manifold, and : ! a bundle of Lie groups over . A dierential
-form on with values in : ! is a law which to any innitesimal simplex
( 0 . . . k ) in associates an element ( 0 . . . k ) in x0 (the bre over 0), in such
a way that if the simplex is degenerate (two vertices equal), then the value of is , the
neutral element in x0 .
We have that -forms are \alternating" in the restricted sense that swapping two of
the vertices i and j for 6= and 1 in the simplex implies a \sign change":
( 0 . . . i . . . j . . . k ) = ( 0 . . . j . . . i . . . k );1
(7)
The result of swapping 0 with i for 1 cannot be compared to the original value,
since these lie in di erent bres, in general. If the group bundle is \constant", i.e. of form
! for some Lie group , forms are also alternating in the sense of swapping
0 with i implies a sign change. This case is of course the same as the case of di erential
forms with values in the group . The proof of the \restricted" case is identical to the one
given above for Proposition 3.1. The sense in which is also changes sign when swapping
0 and i is dealt with in Section 14.
There is an evident way to multiply together two -forms with values in a group
bundle:
( )( 0 . . . k ) := ( 0 . . . k ) ( 0 . . . k )
The set of -forms in fact becomes a group k ( ) under this multiplication the neutral
element is the \zero form" given by ( 0 . . . k ) = x0 .
M
p
k
M
x
x
E
p
M
E
M
M
M
!
! x
k
x
E
x
!
E
k
x
x
! x
i
j
x
i j
x
x
G
x
M
x
x
M
! x
x
x
x
i
G
x
G
!
x
x
k
!
x
x
! x
k
x
x
E
z
z x
x
e
x
:
e
Theory and Applications of Categories,
Vol. 2, No. 7
77
Let be a Lie group. For -valued 1-forms , there is an evident notion of its
coboundary , which is a -valued 2-form:
( 0 1 2) := ( 0 1) ( 1 2) ( 2 0)
(8)
which we may think of as the \curve integral of around the boundary of the innitesimal
simplex ( 0 1 2)". (Warning: it di ers, for non-commutative , from the coboundary
of the Lie-algebra-valued 1-form to which it gives rise, by a correction term ^ we
return to this.)
Using that ( 2 0) = ( 0 2);1, and that an even permutation on the three arguments of does not change the value (since is alternating), we may rewrite the
denition (8) into
( 0 1 2) = ( 1 2) ( 0 2);1 ( 0 1)
(9)
For -valued -forms ( 2), it is also possible, but less evident, to dene its
cobundary +1-form it is less evident in the sense that there is no geometrically compelling way for the order in which to multiply together those factors ( 0 . . . ^ . . . +1),
whose product should constitute ( . . . +1), but it turns out (with proof in the spirit of Section 2) that these factors actually commute (for 2 !), so that the question of
order is irrelevant. We shall consider such \non-commutative" coboundary in Section 11.
For forms with values in a general group bundle ! , one needs one further
piece of structure in order to dene , namely a connection in ! . This is the topic
of the next section.
G
G
d!
!
G
d! x
x
x
! x
x
! x
x
! x
x
!
x
x
x
G
!
! x
x
! x
!
x
d!
d!
d! x
G
k
!
k
x
x
! x
x
! x
x
! x
x
:
k
d!
! x
d! xo
xi
xk
xk
k
!
E
d!
M
E
M
8. Connections
There!are two ways to dene connections, both of which are in terms of the re#exive graph
, the rst neighbourhood of the diagonal one is as an action of this graph on
(1) !
a bundle ! , the other is as a morphism of (re#exive) graphs (1) ! , where is
a groupoid with for its object manifold. The former can be reduced to the latter, by
taking to be the groupoid of isomorphism from one bre of ! to another one.
Here, we temporarily put the emphasis on the latter, groupoid theoretic, viewpoint (due
to Ehresmann).
So, a connection in a groupoid over is a homorphism of re#exive graphs r :
(1) ! , where
(1) denotes the rst neighbourhood of the diagonal. Equivalently, r
is a law which to any innitesimal 1-simplex ( ) associates an arrow r : ! in ,
in such a way that r is the identity arrow at . We shall assume throughout that
is a Lie groupoid, in the well known sense, cf Mac87]. The additive group bundle of a
vector bundle ! , as well as the groupoid GL( ) of a vector bundle considered in
Section 10, are Lie groupoids. The gauge group bundle of a Lie groupoid is a bundle of
Lie groups. Just as for forms, we may then prove that
r = r;1
(10)
One may also write r( ) instead of r .
M
M
E
M
M
M
E
M
M
M
M
x y
x
xx
E
yx
M
E
xy
y x
yx
yx
:
x
y
Theory and Applications of Categories,
Vol. 2, No. 7
78
8.1. Proposition. The set of connections in carries a canonical left action by the
group = 1(
) of 1-forms with values in the gauge group bundle of , and with
this action, it is a translation space over .
(The sense of the term 'translation space' will be apparent from the proof.)
Proof. Given a connection r in , and a 1-form with values in the gauge group
bundle, we get the connection r by putting
( r)( ) := ( )
r( )
(11)
(Note that ( ) is an endo-arrow at , so that the composition here does make sense.)
Given two connections r and ; in , their \di erence" ;
r;1 is the gauge-group-bundle
valued 1-form given by
(;
r;1)( ) := ;( )
r( ) = ;( )
(r( ));1
(12)
Then clearly
(;
r;1) r = ;
and it is unique with this property. The verications are trivial.
Remark. For drawings, the viewpoint of a connection r as an action by the graph (1)
on a bundle ! is usually more appropriate. If r and ; are two such, we present
here a drawing, which exhibits the action (via r and ;) of
on suitable points in ,
as well as the action of the \di erence" 1-form , (which now takes values in the group
bundle of automorphism groups of the bres):
C
G
gauge
G
!
!
!
x y
! x y
! x y
x y :
x
x y
x y
y x
x y
x y
:
M
E
M
x
y
E
!
HHrHyx
HH
( )
Hj
)?;xy
! x y
x
y
9. Combinatorial Bianchi identity
Given a connection r in a Lie groupoid , we dene its curvature as a 2-form with
values in the gauge group bundle of :
(
) = r( )
r( )
r( )
(13)
for (
) an innitesimal 2-simplex. Note that this is going around the boundary of
the innitesimal simplex (
) (in reverse order). We shall adopt a notational shortcut
in the proof of the following formula, by writing for r( ) : ! . Also, we omit
commas upper left indices denote conjugation. Then we have the combinatorial Bianchi
identity (essentially a form of the Homotopy Addition Lemma, cf. the Introduction):
R
R x y z
x y
y z
z x :
yx
y x
x y z
x y z
x
y
Theory and Applications of Categories,
Vol. 2, No. 7
79
9.1. Theorem. Let r be a connection in a groupoid , and let be its curvature. Then
for any innitesimal 3-simplex (
),
xy
)
( )
( )
( )
x= (
We shall below interpret the expression here as the coboundary or covariant derivative
of in a \complex" of group-bundle valued forms, with respect to r. This is still a
purely \combinatorial" gadget, but we shall later specialize to the case of the general
linear groupoid of a vector bundle with connection, and see that our formulation contains
the classical one.
R
x y z u
id
R yzu
R xyu
R xuz
R xzy :
R
Proof. With the streamlined notation mentioned, the identity to be proved is
xy
(
)
(
)
(
)
(
) = x
yz
zu
uy
yx
xy
yu
ux
xu
uz
zx
xz
zy
yx
id
the proof is now simply repeated cancellation: rst remove all parentheses, then keep
cancelling anything that occurs, or is created, which is of the form
etc., using (10)
one ends up with nothing, i.e. the identity arrow at .
The reader may like to see the geometry of this proof by drawing a tetrahedron with
vertices named
and then trace that path (of length 14 units) which the left hand
side of the above identity denotes cf. the description in the Introduction.
yx
xy
x
x y z u
10. Linear connections
We shall consider in particular linear connections in vector bundles ! they are
connections in the above groupoid theoretic sense, if we take the groupoid to be the
groupoid GL( ) of linear isomorphisms between the bres of ! . So for
in
, rxy : y ! x is a linear isomorphism, and rxx is the identity map of x. We then
have the following comparison, which we shall use later. Note that the rst term of the
left hand side is just the curvature (
) 2 ( x), applied to ( ). Since this is
2 ( x ), the left hand side itself is (log )(
), applied to ( ).
10.1. Proposition. Let be a section of the vector bundle ! , and let r be a
linear connection on it. Then for any innitesimal 2-simplex
rxy ryz rzx ( ) ; ( ) = rxy ryz ( ) ; rxz ( )
(14)
E
E
M
E
E
M
E
x
y
E
R x y z
e
M
GL E
GL E
R
f x
x y z
f x
f
E
M
x y z
f x
f x
f z
f z :
Proof. Since the question is local, we may assume that the bundle is V M
! for a
vector space , and we may assume that = Rm . Then there is a 1-form on with
values in ( ) such that
rxy ( ) = ( ( )( ) )
for all
2 and 2 . Now given a 2-simplex
and a section , there is a
unique linear map : = Rm ! such that ( ) = ( ) + ( ; ) for all 2 M1( ).
V
M
M
M
GL V
v y
x
y
M
v
g
M
x y
V
v
x
x y z
V
f u
f x
f
g u
x
u
x
Theory and Applications of Categories,
Vol. 2, No. 7
80
Rewriting r in terms of , the left hand side of (14) is simply (
)( ( )) ; ( ).
Writing ( ) = ( ) + ( ; ) and using linearity of ( ) : ! for any
, we
calulate the right hand side to be
( ) ( ) ( ); ( ) ( )+ ( ) ( ) ( ; ); ( ) ( ; )
(15)
Now since the values of are in ( )
( ) which is a vector space, there is
m
( ) such that ( ) = V + x( ; ). In particular,
a linear map x : R !
( ) ( ; ) = ( ; ) + x( ; ) ( ; ), and the last term here vanishes, because
it comes from a bilinear expression with ;
0 substituted in two places. So the last
term in our expression (15) is just ; ( ; ). One may similarly see that the second but
last term is ( ; ) (rewrite ( ; ) as ( ; )+ ( ; )). So the two last terms cancel.
Therefore (14) will follow if we prove in the ring
( ) that
( ) ( ) ( ); = ( ) ( ); ( )
(16)
Both sides clearly give 0 if the 2-simplex is degenerate, so both dene 2-forms with values
in (the additive group of)
( ). In particular, let denote the right hand side, so
(
)= ( ) ( ); ( )
.
Also, there is an
( ) valued 1-form on such that ( ) = + ( ), for
Then ^ is an
( )-valued 3-form, and therefore
0 = ( ^ )(
)= ( ) (
)
and hence
( )(
)= (
)
So we may multiply the expression dening (
) on the left by ( ) without changing the value. So the right hand side of (16) equals
( ) ( ) ( );
(17)
But the expression ( ) ( ) ( ) is a 2-form (
), hence alternating, so that
we may perform a cyclic permutation of the arguments
. Applying it to (17) gives
the left hand side of (16), proving the Proposition.
More generally, we may replace the section in the above Proposition by an -valued
-form, for any :
10.2. Proposition. Let ! be a vector bundle equipped with a connection r.
Let be an -form on with values in . Then for any innitesimal + 2-simplex
1 . . . n , we have
rxy ryz rzx ( 1 . . . n) ; ( 1 . . . n)
= rxy ryz ( 1 . . . n) ; rxz ( 1 . . . n)
f z
f x
d x y z
g z
x y y z f x
x
u v
x z f x
x z g z
x y y z g z
GL V
g z
x
x u
z
x g z
g z
x
g z
x
u
x z g z
v
x :
id
u
x
x
z
g z
x
V
f x
E nd E
E nd V
x
V
f x
x
x
g z
y
g y
x
E nd V
x y y z z x
id
x y y z
E nd V
x y z
x y y z
E nd V
x z :
x z :
M
u v
id
u v
u
E nd V
x z y z
x z x y z
z x x y z
x y z :
x y z
z x x y y z
z x x y y z
z x
id:
d x y z
x y z
f
n
E
n
E
!
x y z z
n
M
M
E
n
z
! x z
! z z
z
z
! x z
! z z
z
z
:
v
Theory and Applications of Categories,
Vol. 2, No. 7
81
Proof. The proof is the same as that of the previous Proposition, except that the linear
function g appearing there, and which is the degree 1 term of the Taylor series for the
map f : Rm ! V , is replaced by a linear map Rm ! Altn(Rm V ), again the degree 1
term of the Taylor series at x for !, viewed as a map from M = Rm to the vector space
Altn(Rm V ) of n-linear alternating maps from Rm to V .
11. Covariant derivative
We consider here a group bundle E ! M (typically the additive group bundle of a vector
bundle). Also, let there be given a connection r on it, i.e. a connection in the groupoid
GL(E ), or equivalently, an action by the neigbourhood graph M(1) !
! Mk on E ! M ,
acting by group homomorphisms. As in Section 7, we consider the group (E ) of k-forms
with values in E . We shall dene a map (not a group homomorphism, unless the group
bundle is commutative)
dr : k (E ) ! k+1 (E ):
Let ! 2 k (E ), and let (x0 . . . xk+1) be an innitesimal k + 1-simplex in M . As
usual, @i of such a simplex denotes the one obtained by omitting the i'th vertex. Then
we put
(dr !)(x0 . . . xk+1) = r(x0 x1)!(x1 . . . xk+1)
Y !(@i(x
k+1
1
0
. . . xk+1)1
where the sign in the exponent is ; if i is odd, + if i is even. (It turns out that for k 2,
the factors commute (essentially by the arguments at the end of Section 2), so that their
order is irrelevant.) For k = 0, and f 2 0(E ) (so f is a section of E ), dr (f ) is the
covariant derivative of f with respect to r,
dr (f )(x y) = (r(xy)f (y)) f (x);1:
(18)
If we consider a connection r in a Lie groupoid !
! M , we get a connection in
the group bundle gauge( ), since acts on gauge( ) by conjugation. We denote this
connection adr.
We now have the following reformulation of the combinatorial Bianchi identity (Theorem 9.1):
11.1. Theorem. Let r be a connection in a Lie groupoid !
! M , and let R be its
2
ad
r
curvature, R 2 (gauge( )). Then d (R) is the \zero" 3-form, i.e. takes only the
neutral group elements in the bres as values.
Proof. Let x y z u form an innitesimal 3-simplex. We have by denition of dadr that
(dadr(R))(xyzu) = adrxy R(yzu)
R(xzu);1
R(xyu)
R(xyz);1
Theory and Applications of Categories,
Vol. 2, No. 7
82
(omitting commas for ease of reading). Now the two middle terms may be interchanged,
by arguments as those of Section 2. We then get the expression in the combinatorial
Bianchi identity in Theorem 9.1, and by the Theorem, it has value x.
It is not true that r
r is the \zero" map, unless the curvature of the connection
vanishes see Section 13 below. To make the comparison with classical curvature, we
calculate here r
r : 0( ) ! 2( ) for the case of a commutative group bundle
! (which we write additively):
11.2. Proposition. Let 2 0( ), so is a section of the bundle , and let (
)
be an innitesimal 2-simplex. Then
id
d
d
E
d
d
E
E
M
f
E
r (dr (f )) (x
d
f
y z
E
x y z
) = rxy ryz ( ) ; rxz ( )
f z
f z :
For an -valued 1-form, r (
) = ( ) ; ( ) + rxy ( ). Now
let be given by the expression (18), but written additively, so ( ) = r ( ) =
rxy ( ) ; ( ) then we get, using additivity of rxy , six terms, four of which cancel, and
the two remaining ones give the expression claimed.
We recognize the right hand side here as the one we have in Proposition 10.1. So we
get by combining Propositions 10.1 and 11.2:
11.3. Proposition. For a section of the vector bundle with a linear connection r,
we have
(log )(
)( ( )) = ( r r )(
)
where is the combinatorial curvature of r, 2 2 ( ( )).
The \classical" curvature of a connection r in a vector bundle is usually dened in
terms of the right hand side of this equation. So the Proposition establishes the comparison that the logarithm of the combinatorial curvature equals the classical curvature.
Proof.
!
E
d
! x y z
! x y
! x z
!
! y z
! x y
f y
d
f
x y
f x
f
E
R
x y z
f x
d
R
d
f
R
x y z
GL E
12. Classical Bianchi Identity
We consider a vector bundle ! equipped with a linear connection r. Then we
get induced connections r in the group-, resp. ring-bundle ( ),
( ). We shall
consider diagrams, for 1,
E
M
ad
GL E
k
(
k
( ))
GL E
adr
d
>
k+1(
k (log)
(
k
_
( ))
GL E
k+1 (log)
( ))
E nd E
>
adr
d
(
k+1
_
( ))
E nd E
:
E nd E
Theory and Applications of Categories,
Vol. 2, No. 7
83
Even though, for 2 , log : ( x ) !
( x) is only partially dened (namely
on the respective monads M1( x), at least), the vertical maps are globally dened, since
for any form 2 k ( ( )), its values are
x . We shall address the question of
commutativity of these diagrams. The (partial) map log is equivariant with respect to the
connections r in ( ) and
( ), but it is not a group homomorphism ( x) !
( x), not even on M1( x) due to this, there is no \cheap" reason why these diagrams
should commute. However, log has the homomorphism property with respect to pairs of
commuting elements in ( x ) since the factors that dene adr for ( ) do commute
for 2, as stated also in Section 11, we conclude that the square above actually does
commute for 2.
Using the commutativity for = 2, and the combinatorial Bianchi identity (in the
form of Theorem 11.1), we shall prove
12.1. Theorem. (Classical Bianchi Identity) Let denote the curvature of a linear connection r in a vector bundle ! . As an element 2 2 ( ( )), adr ( ) is 0.
Proof. Consider the combinatorial curvature 2 2 ( ( )) of the connection r. By
the combinatorial Bianchi Identity, in the guise of Theorem 11.1, it goes to the \zero"
form by adr, hence also to the zero form by 3(log)
adr. Chasing the other way
round in the square rst gives log , which by Proposition 11.3 is the classical curvature
of r, so from the commutativity of the square follows adr( ) = 0.
For = 1, the square does not commute, in general.This fact is related to the MaurerCartan formula. For the case of a constant Lie group bundle, this relationship was considered synthetically in Koc82]. Here we do the more general case of a group bundle but, on the other hand, it is more special in another direction, since it deals only with a
group bundle of form ( ).
12.2. Proposition. Let 2 1( ( )), where ! is a vector bundle with a
connection r. Then
log( adr ( )) = adr(log ) + log ^ log
x
M
GL E
E nd E
e
!
ad
GL E
e
GL E
E nd E
E nd E
GL E
e
GL E
d
!
GL E
k
k
k
R
E
R
M
R
E nd E
d
R
GL E
d
d
R
R
R
d
R
k
GL E
!
GL E
d
!
E
d
!
M
!
!:
(The wedge here is with respect to the multiplication
in the ring
( ), which is
non-commutative so there is no reason for the wedge of a 1-form with itself to vanish.)
Proof. Let us write ( ) = x + ( ), so = log 2 1 (
( )). Let us write
conjugation by r( ) by an upper left index . Then for an innitesimal 2-simplex
, we have
r (
) =xy ( y + ( ))
( x ; ( ))
( x + ( ))
E nd E
! x y
e
x y
x y
!
E nd E
xy
x y z
d
! x y z
Multiplying out, we get
e
y z
e
x z
e
x y
+xy ( ) ; ( ) + ( )+
)
( ) +xy ( )
( ) ; ( )
(
ex
;xy (
yz
x z
yz
x z
y z
x y
x y
x z
x y
)
:
Theory and Applications of Categories,
Vol. 2, No. 7
84
plus a threefold product, which is easily seen to vanish (e.g. by coordinate calculations
like the following). The rst line in the formula is ( x + r )(
). We prove that
the second line equals ( )
( ) = ( ^ )(
). We claim that each of the three
terms in the second line give plus or minus ( ^ )(
) (twice plus, once minus). Let
us consider the rst only, the two others are similar. Since the question is local, we may
assume that = for a vector space, and that = Rm and we may assume
that ( ) = ( ; ) for
, with : !
( ) linear in the second
variable. Let = + 1 and = + 2. Then
e
x y
x z
E
u v
M
V
f u v
y
v
z
xy (y
z
x y z
x y z
M
u
d
x y z
V
u
x
d
f
x
)
(
x z
M
M
E nd V
d1 d2
; 1)
(
d
) =xy ( +
f x
f x d2
d
)
expanding the rst term out by linearity in the second variable of , we get two terms,
one of which vanishes because of \bilinear occurrence" of 2, and we are left with xy ( +
in its rst variable, and get xy ( ; 1)
1 ; 1)
(
2 ). Now we Taylor expand
( 2) plus a term which vanishes because of \bilinear occurrence" of 1. Also, since we
are conjugating with r( ) 2 ( ), whose logarithm depends linearily on ; = 1,
( ; 1) is xed under this conjugation. We are left with
f
d
d
d
f x d
f x
f
f x
f x d
x y
f x
d
d
GL V
y
x
d
d
( ; 1)
(
f x
d
f x d2
)=; (
f x d1
)
(
f x d2
)=; (
x y
)
(
x z
) = ;( ^ )(
)
x y z :
All said, we conclude that
adr ! (x
d
y z
)= x+ (
e
d x y z
) + ( ^ )(
x y z
)
where = log( ). Now subtracting x from the right hand side of this expression gives
something which is 0 2
( x) (in coordinates, it depends linearily on ; , say)
and this something is therefore the logarithm. This proves the Proposition.
!
e
E nd E
13. Analyzing dr
y
x
dr
We sketch in this section how the curvature enters in describing how the composite r
r
fails to be the zero map in the complex of -valued forms (where ! is a vector
bundle equipped with a linear connection r). This is classical, and of importance in
constructing the characteristic classes for the bundle ! .
Since the ring bundle ev ( ) ! acts in a bilinear way on ! simply by
evaluation
( ) M ! , the wedge product ^ is a well dened -valued
( )-valued 2-form, say
+ 2 form whenever is an -valued -form, and an
the (classical) curvature of the connection r. This is what denotes in the following
Proposition.
13.1. Proposition. Let be an -valued -form in the vector bundle . Then
d
E
E
E
E nd E
E nd E
k
E
M
M
E
R
E
!
E
M
R
k
M
!
E
E nd E
R
!
E
k
r (dr (! )) = R ^ !:
d
E
d
Theory and Applications of Categories,
Vol. 2, No. 7
85
We shall do the case = 1 only. The calculation is much similar to the
one in Proposition 11.2 (for any 1, in fact). For = 1, we get 12 terms in
r ( r ( ))(
), where (
) is an innitesimal 3-simplex. These terms are
Proof.
k
k
d
d
!
x y z u
k
x y z u
rxy(ryz ( ) ; ( ) + ( ))
;(rxz ( ) ; ( ) + ( ))
+rxy ( ) ; ( ) + ( )
;(rxy ( ) ; ( ) + ( ))
(in the rst line, use linearity of rxy to get three terms). Ten of these twelve terms cancel
! z u
in pairs, and we are left with
! y u
! y z
! z u
! x u
! x z
! y u
! x u
! x y
! y z
! x z
! x y
rxy(ryz
) ; rxz (
(
! z u
This equals, by Proposition 10.2, (with
rxyryz rzx
z1
)
! z u :
= )
(
u
); (
! x u
)
! x u :
The rst term here is the combinatorial curvature (
) of r, applied to ( ), so
)
when we subtract ( ), we obtain its logarithm, i.e. the classical curvature (
(applied to ( )). But by the denition (5) of wedge, (with evaluation
( ) M !
as the bilinear map),
R x y z
! x u
R x y z
! x u
! x u
E nd E
E
E
(
R x y z
)( (
)) = (
! x u
R
^
!
)(
)
x y z u :
14. Curvature dierence
We consider in this section a Lie groupoid !
! . When we have two connections;r1
and ; in , we may form their \di erence", which is a
( )-valued 1-form ;
r ,
cf. Section 8. So (;
r;1)( ) = ;xy
ryx. Also, we have two
( )-valued 2-forms,
r
;
namely their curvatures and .
14.1. Proposition. We have
adr (;
r;1) = ( r );1
;
(19)
M
gauge
x y
R
gauge
R
d
R
R :
We rst prove a Lemma which in some sense expresses how certain bundle valued
forms also are alternating with respect to interchanging the rst vertex with another one:
Proof.
Theory and Applications of Categories,
Vol. 2, No. 7
86
14.2. Lemma. Let be a 2-form with values in the gauge group bundle of a Lie groupoid
. For any connection r in , we have
r( ) (
)= (
)
The similar result holds for any -form with 1, using cyclic permutation of the
vertices, and inserting the exponent (;1) .
z x
x y z
k
z x y :
k
k
Proof. Since a Lie groupoid is locally trivial (cf. Mac87]) and the question is local, we
may assume that =
2-form ,
M
G
M
for a Lie group , and that is given by a -valued
G
G
(
)=( (
) )2
and also that r is given by a -valued 1-form
r =( ( ) )
Then the question reduces to whether (
) (= (
), since is alternating by
Proposition 3.1) is xed by conjugation by ( ), i.e. whether (
) and ( )
commute. This is so, by arguments similar to those of Section 2 (bilinearity in 2 =
; ).
To prove (19), let
form an innitesimal 2-simplex. We want to prove (omitting
commas for ease of reading)
r (;
r;1 )( ) = ( r ( ));1
; ( )
We rewrite the left hand side, by using the Lemma twice (rst for the connection ;, then
for the connection r) and get
;( )r( )
r (;
r;1 )( )
where upper left index denotes \conjugation by". Now, expanding the left hand side into
its constituents, using the denition of covariant derivative r , we get
; r ((r (; r )r )(r ; )(; r ))r ;
where the parentheses are just meant as an aid for the checking of the correctness of the
expansion. Removing all parentheses, and cancelling anything of the form r r (and
similarly for ;) that occurs or is created, we end up with
r r r ; ; ; = r ( ) ;( ) = ( r ( ));1 ; ( )
This proves the Proposition.
We shall use this Proposition in order to prove that the trace of r (and its wedge
powers) only depends on r up to a coboundary, so its deRham cohomology class does
not depend on the choice of connection.
x y z
x x y z
G
x
M
G
M
!
x ! x y
xy
y :
x y z
z x y
! x z
x y z
! x z
d
z
x
z x y
ad
d
zxy
zy
yx
R
zxy
ad
d
R
zxy :
xyz
ad
d
zy
yx
xy
yz
zy
yx
xz
zx
xy
yx
xy
yz
yx
zy
yx
xz
zx
xy
yz
R
zyx R
zxy
R
zxy
R
R
zxy :
xy
Theory and Applications of Categories,
Vol. 2, No. 7
87
15. Characteristic classes
We sketch how the theory of connections in vector bundles leads to characteristic classes.
This is the classical Chern-Weil theory, and there is not much specically synthetic about
the way we present it here.
For a vector bundle ! , we have for each 2 the linear trace map
E
M
x
M
( )!R
Trace :
E nd Ex
and collectively, they dene a linear map of vector bundles
Since for any linear isomorphism : x ! y , and any
Trace (
;1), it follows that the vector bundle map
a
E
E
( ) ! R over .
2 ( x), Trace ( ) =
E nd E
a
( )!
E nd E
M
E nd E
Trace :
M
a
R
M
is equivariant for the groupoid = GL( ) (with the groupoid acting by conjugation on
( ), and trivially on R). In particular, if r is a linear connection in ! ,
we get homomorphisms
E
E nd E
M
E
( )) ! k (
Trace : k (
E nd E
M
R) = (
k
M
M
)
(20)
commuting with the 's (respectively adr and the usual deRham ).
From the Bianchi identity, in the guise of Theorem 11.1, we therefore get that the
trace of the (classical) curvature of r is a cocycle in 2( ).
Since Trace :
( x ) ! R does not preserve the multiplicative structure, the
induced maps (20) will not preserve wedge products, in general. However, let 2
1( ( )). Then
Trace ( ^ ) = 0
For the left hand side, applied to an innitesimal 2-simples
, yields
d
d
d
R
M
E nd E
!
E nd E
!
!
:
x y z
Trace (
!
^
!
)(
x y z
) = Trace ( (
! x y
)
(
! x z
)) = Trace ( (
! x z
(the last equality by the fundamental property of trace that Trace (
and then we continue the equation
= Trace ((
!
^
!
)(
x z y
)) = ;Trace ((
!
^
!
)(
a
)
(
! x y
))
) = Trace (
x y z
b
b
)),
a
))
since forms are alternating. But then the total equation implies that Trace ( ^ )(
)=
0.
15.1. Theorem. Let r and ; ber two linear
connections on the vector bundle !
, with (classical) curvatures
and ; , respectively. Then the deRham cocycles
Trace ( r ) and Trace ( ; ) 2 2( ) dene the same cohomology class.
!
!
x y z
E
M
R
R
R
M
R
Theory and Applications of Categories,
Vol. 2, No. 7
88
Let = ;
r;1 be the \di erence" 1-form of the two connections, viewed as
connections with values in the groupoid GL( ). Thus, for the combinatorial curvatures
r and ; , we have, by Proposition 14.1,
Proof.
!
E
R
R
adr
d
( ) = ( r );1
!
R
R
;
now apply log the values of r and ; commute, as observed in Section 12, so that log
has the homom
COMBINATORICS OF CURVATURE, AND THE BIANCHI
IDENTITY
ANDERS KOCK
Transmitted by F. William Lawvere
ABSTRACT. We analyze the Bianchi Identity as an instance of a basic fact of combinatorial groupoid theory, related to the Homotopy Addition Lemma. Here it becomes
formulated in terms of 2-forms with values in the gauge group bundle of a groupoid,
and leads in particular to the (Chern-Weil) construction of characteristic classes. The
method is that of synthetic di erential geometry, using \the rst neighbourhood of the
diagonal" of a manifold as its basic combinatorial structure. We introduce as a tool a
new and simple description of wedge (= exterior) products of di erential forms in this
context.
Introduction
We shall give a proof of the Bianchi Identity in di erential geometry. This is an old
identity the novelty of the proof we present is that it derives the identity from a basic
identity in combinatorial groupoid theory, and permits rigorous pictures to be drawn of
the mathematical objects (connections, curvature,...) that enter.
The method making this possible here, is that of synthetic di erential geometry (\SDG" see e.g. Koc81]), which has been around for more than 20 years but in contrast
to most of the published work which utilizes this method, the basic notion in the present
paper is the rst order notion of neighbour elements x y in a manifold M , rather than
the notion of tangent vector D ! M (which from the viewpoint of logic is a second order
notion). This latter notion really belongs to the richer world of kinematics, rather than
to geometry: tangent vectors are really motions.
In so far as the the specic theory of connections is concerned, it has been dealt with
synthetically both from the viewpoint of neighbours, and from the viewpoint of tangents
for the latter, cf. KR79], MR91], and more recently and completely, Lav96], (which also
contains a proof of the Bianchi identity, but in a completely di erent spirit).
On the other hand, the theory of connections, and the closely related theory of differential forms, has only been expounded from the combinatorial/geometric (neighbour-)
viewpoint in early unpublished work by Joyal, and by myself, Koc80], Koc83], Koc82],
Koc81] (section I.18), Koc85] but it has only been mentioned in passing in the monographs MR91] (p. 384), Lav96] (p. 139 and 180).
Received by the editors 20 June 1996 and, in revised form, 6 September 1996.
Published on 26 September 1996
1991 Mathematics Subject Classication : 58A03, 53C05, 18F15 .
Key words and phrases: Connection, curvature, groupoid, rst neighbourhood of the diagonal.
c Anders Kock 1996. Permission to copy for private use granted.
69
Theory and Applications of Categories,
Vol. 2, No. 7
70
My contention is that the combinatorial/geometric viewpoint comes closer to the geometry of some situations, and permits more pictures to be drawn. One may compare
the theory of di erential forms, as expounded here, (and in Koc82] and Koc81] Section
I.18.)), with the theory of di erential forms as seen from the viewpoint of tangents, and
expounded synthetically in KRV80], Koc81] Section I.14, MR91], Min88], Lav96].
Thus, the fact that di erential k-forms (k 2) should be alternating is something that
is deduced in the combinatorial/geometric approach, but is postulated in the more standard
synthetic approach (as in Koc81], Def. I.14.2). Likewise, the denition of wedge product
of forms as presented in the present paper from the combinatorial/geometric viewpoint
(for the rst time, it seems), is more self-explanatory than the theory of Min88] and
Lav96] p. 123 (their theory is here anyway quite close to the \classical" formalism).
The piece of drawable geometry, which, in the approach here is the main fact behind
the Bianchi identity, is the following: consider a tetrahedron. Then the four triangular
boundary loops compose (taken in a suitable order) to a loop which is null-homotopic inside
the 1-skeleton of the tetrahedron (the loop opposite the rst vertex should be \conjugated"
back to the rst vertex by an edge, in order to be composable with the other three loops).
- This fact is stated and proved in Theorem 9.1 (in the context of combinatorial groupoid
theory). Ronald Brown has pointed out to me that this is the Homotopy Addition Lemma,
in one of its forms, cf. e.g. BH82] (notably their Proposition 2).
In so far as the di erential-geometric substance of the paper is is concerned (i.e.
leaving the question of method and pictures aside), the content of the paper is part of the
classical theory of connections in vector bundles, leading to the Chern-Weil construction
of characteristic classes for vector bundles. I learned this material from Mad88] Section
11, (except for the groupoids, which in this context are from Ehresmann) cf. also MS74].
One cannot say that I chose to couch this theory into synthetic terms this is not a matter
of choice, but rather a necessity to see what connection theory means in terms of the
neighbour relation, now that this relation has forced itself onto our minds.
I would like to acknowledge some correspondence with Professor van Est in 1991, where
he suggested a relationship between combinatorial groupoid theory and the neighbourhood
combinatorics of synthetic di erential geometry. He also sent me his unpublished vE76],
where in particular some di erential-geometric terminology (curvature, Bianchi identity,..)
is used for notions in combinatorial groupoid theory.
1. Preliminaries
We shall need to recall a few notions from synthetic di erential geometry. The main thing
is that for any manifold M , there is, for each integer k, a notion of when two elements
x y 2 M are k-neighbours, denoted x k y. In particular, x 1 y is denoted x y. (The
\set" of pairs (x y) 2 M M with x y is what in algebraic geometry is called the rst
neighbourhood of the diagonal, or M(1) if M is an ane scheme Spec(A), then M M
is Spec(A A) and M(1) = Spec((A A)=I 2) where I is the kernel of the multiplication
map A A ! A.) There exist categories containing the category of smooth manifolds as
Theory and Applications of Categories,
Vol. 2, No. 7
71
a full subcategory, but in which any manifold acquires such relations like and k
as new subobjects of one talks about these new (sub-)objects in terms of their
\elements", although (1), say, does not have any more points than itself does.
The set of -neighbours of an element 2 is denoted Mk ( ) and called the -monad
around . The union of all the -monads around is denoted M1( ).
In particular, the -monad around 0 2 Rn , i.e. the \set" (object) of -neighbours of
0 2 Rn is denoted k ( ) if = 1, we just write ( ). So
M
M
M
M
M
k
x
x
M
k
x
x
k
x
k
D
k
n
k
D n
M1(0) = ( )
D n :
( ) Rn as the set
One may also describe
Dk n
f( 1 . . .
d
dn
) j the product of any + 1 of the i's is 0g
k
d
:
Let us explicitly note the following property of ( ): If any multilinear function has
an element from ( ) as argument in two di erent places, then the value is zero.
The sub\set" k ( ) Rn is one of the important kinds of \innitesimal" objects in
SDG. Another one is k ( ) Rn . . . Rn ( times) which we shall describe now.
First: an innitesimal -simplex in a manifold is a +1-tuple of elements ( 0 . . . k ),
such that for any , i j . Such a simplex is called degenerate if two of the vertices
i and j are equal ( 6= ).
A -tuple of elements ( 1 . . . k ) in Rn is said to belong to k ( ) if the + 1-tuple
(0 1 . . . k ) is an innitesimal -simplex so not only are all i in ( ), but also each
n
n
i ; j . This innitesimal subobject of R . . . R ( times) was introduced in Koc81]
there it was denoted ~ ( ).
From Koc81] Section I.18, we quote the following result, which seems not to have
been considered in any of the subsequent treatises on SDG, but which is crucial to the
following.
1.1. Theorem. Any map : k ( ) ! R which for each = 1 . . . has the property
that ( 1 . . . k ) = 0 if i is 0, is the restriction of a unique -linear alternating map
: (Rn )k ! R.
From the Theorem, we may immediately deduce a more general one, where the
codomain R is replaced by any other nite dimensional vector space .
D n
D n
D
n
D n
k
k
i j
x
x
i
k
d
d
M
x
k
x
x
x
j
d
d
d
D n
k
d
d
k
D n
k
D k n
!
! d
d
D n
i
d
k
k
!
V
2. Logarithms on General Linear Groups
Let
be a nite dimensional vector space (of dimension , say). Then we have the ring
( ), whose additive group is a vector space of dimension 2, and whose group of units
is a Lie group, denoted ( )
( ). The identity element 2 ( ) is denoted .
If k 0 in
( ), it follows by matrix calculations (identifying with Rq , and
( ) with the vector
space
of matrices) that k+1 is the zero endomorphism, so
P
1
ap
the exponential series 0 p! has only nitely many non-zero terms, and it is actually an
automorphism ! (due to the rst term which is ) it is thus an element of ( ),
V
q
E nd V
q
GL V
a
E nd V
GL V
E nd V
E nd V
V
q
V
V
e
q
a
e
GL V
Theory and Applications of Categories,
Vol. 2, No. 7
72
and it is denoted exp(a). Similarly, if b k e2 in GL(V3 ), log(b) 2 End(V ) is dened by
the logarithmic series log(b) = (b ; e) ; (b;2e) + (b;3e) ; . . ., which likewise terminates.
The fact that exp and log establish mutually inverse bijections Mk (0) = Mk (e) for all k,
and hence M1(0) = M1(e), is the standard power series calculation. Likewise, if a and
b are commuting endomorphisms in M1 (0) End(V ), the formula
exp(a + b) = exp(a) exp(b)
(1)
comes by a standard calculation with series. Similarly for log on commuting elements in
End(V ).
In particular, if V is 1-dimensional, we get the standard homomorphisms exp : (R +) !
(R ) and log (natural logarithm), which here happen to be extendible beyond the respective 1-monads, and also are group homomorphisms.
The following is a basic fact in Lie theory:
2.1. Proposition. Let and be e 2 GL(V ). Then the logarithm of the group
theoretic commutator is the ring theoretic commutator of their logarithms.
Proof. Let a and b be the logarithms of and , respectively, so = e + a. From
e
follows a 0 2 End(V ). Similarly for b. Using coordinate calculations in the algebra of
q q matrices, it is immediate from a 0 that a a = 0, and hence that the multiplicative
inverse of = e + a is e ; a similarly for . Thus
;1 ;1 = (e + a)(e + b)(e ; a)(e ; b)
which multiplies out in End(V ) by distributivity to give 16 terms. Some of these, like aba
contain a repeated a or b factor, and are therefore 0, again by matrix calculations, using
a 0, resp. b 0. The \rst order" tems are a b ;a and ;b, so they cancel. Also aa
and bb vanish, and we are left with e + ab ; ba since ab ; ba 0 (being 0 if a = 0), we
get that log(;1 ;1) = ab ; ba = a b], proving the Proposition.
In particular, 1-neighbours of e in GL(V ) commute if and only if their logarithms do
in End(V ) (this actually also holds for k-neighbours of e). Among pairs which always
commute in End(V ) are pairs of form (d) (d), where and are linear maps Rm !
End(V ) and d 2 D(m). This again follows from matrix calculations.
We shall give a sample of other commutation laws that can be derived from this
principle, and which are to be used later. Assume f and g are bilinear maps Rm Rm !
End(V ), and assume 0 d1 d2 d3 form an innitesimal 3-simplex in Rm. Then f (d1 d2)
commutes with g(d1 d3) as well as with g(d2 ; d1 d3 ; d1). For the former, this follows
by keeping d2 and d3 xed then each of the two elements depend linearily on d1, and
we are back at the previous situation similarly, for the second assertion, we rewrite
g(d2 ; d1 d3 ; d1) as a sum of four terms, using bilinearity of g then each of the four
terms has either a d1 or a d2 factor, and hence commutes with f (d1 d2).
The results of this section generalize to \abstract" Lie groups G, without representing
these as subgroups of some GL(V ) this depends on consideration of fragments of the
Theory and Applications of Categories,
Vol. 2, No. 7
73
Campbell-Baker-Hausdor series. In particular, if exp : D(p) ! M1(e) G is any
bijection taking 0 to e, additive inversion v 7! ;v in D(p) corresponds to multiplicative
inversion in G.
3. Dierential forms with values in Lie groups
Let M be a manifold, and let G be a Lie group. For k = 0 1 . . ., we dene a dierential
k-form with values in G to be a law ! which to each innitesimal k-simplex in M
associates an element !() 2 G the only axiom is that if the simplex is degenerate,
then !() = e, the neutral element of G. (Remark: it even suces to assume that
!() = e for any which is degenerate in the special way that it contains the vertex x0
repeated.)
3.1. Proposition. Let ! be a dierential k-form on a manifold M with values in a Lie
group G. Then ! is alternating, in the sense that swapping two of the vertices in the
simplex implies a \sign change":
!(x0 . . . xi . . . xj . . . xk ) = !(x0 . . . xj . . . xi . . . xk );1
(2)
for i 6= j .
m
Proof. Since the question is local, we may assume that M = R . Also, since values
of di erential forms are always 1-neighbours of the neutral element in the value group
G, only M1(e) G is involved and M1(G) = M1(0) Rn , i.e. M1(e) = D(n),
where n is the dimension of G. Now, (as mentioned above), under this isomorphism, the
inversion g 7! g;1 in M1(e) corresponds to the inversion v 7! ;v in D(n). Therefore,
the problem reduces to the case where M = Rm and G = Rn (under addition). Now any
innitesimal k-simplex (x0 . . . xk ) in Rm may be written (x0 x0 + d1 . . . x0 + dk ), where
(d1 . . . dk ) 2 k D(m). Write
!(x0 x0 + d1 . . . x0 + dk ) = f (x0 d1 . . . dk ):
(3)
Then for each x0, f (x0 ; . . . ;) is a function k (D(m)) ! Rn, (with value zero if one
of the blanks is lled with a zero), and hence, by Theorem 1.1, it extends to a k-linear
alternating function (Rm )k ! Rn, which we also denote f (x0 ; . . . ;). From this, it
is clear that we get a minus on the value of !(x0 x1 . . . xk ) if xi is swapped with xj
for i j 1, since this amounts to swapping di and dj . (For the same reason, it is also
clear that we get 0 if xi = xj for some i 6= j from this, the parenthetical remark prior
to the statement of the Proposition follows.) The argument for the case of swapping x0
with one of the other xi's is a little di erent. Without loss of generality, we may consider
swapping x0 and x1 = x0 + d1. This we do by a Taylor expansion in the rst variable for
the function f : Rm (Rm)k ! Rn , which is multilinear in the last k arguments. The
values to be compared are f (x0 d1 d2 . . . dk ) and f (x0 + d1 ;d1 d2 ; d1 . . . dk ; d1),
and the Taylor development of this latter expression gives
f (x0 ;d1 d2 ; d1 . . . dk ; d1) + D1f (x0 ;d1 d2 ; d1 . . . dk ; d1)(d1)
Theory and Applications of Categories,
Vol. 2, No. 7
74
where D1f denotes the di erential of f in the rst variable it is linear in the original k
variables as well as in the extra variable. Because d1 appears in two places in the k + 1
linear expression D1f , this term vanishes, so we are left with f (x0 ;d1 d2 ;d1 . . . d ;d1)
which we expand into a sum, using its multilinearity. In this sum, all but one term
contain a ;d1 as argument in two di erent places, so vanishes. The remaining term is
f (x0 ;d1 . . . d ) = ;f (x0 d1 . . . d ) = ;!(x0 x1 . . . x ), as desired. This proves the
Proposition.
Remark. It is tempting, (but unjustied without reference to Theorem 1.1), to attempt
to start the argumentation: \keep x2 . . . x xed, and consider !(; ; x2 . . . x ) as a
function of the two variables x0 and x1 = x0 + d1 only as a function f of d1, it extends by
the basic axiom of SDG to a linear map on the whole vector space". This is unjustied
for, it would require that f is already dened on the whole of D(m), but d1 is not free to
range over the whole of D(m), since it is still tied by the condition that x0 + d1 should
be neighbour to all the x2 . . . that we are keeping xed.
The fact that the \alternating" property comes so easily, namely just by proving that
the value on degenerate simplices is trivial, leads to a very simple description of wedge
product of forms this we deal with in Section 5.
There is an evident way to multiply together two k-forms on M with values in the
group G:
(! )(x0 . . . x ) := !(x0 . . . x ) (x0 . . . x )
The set of k-forms in fact becomes a group (M G) under this multiplication the neutral
element is the \zero form" z given by z(x0 . . . x ) = e.
k
k
k
k
k
k
k
k
k
k
k
4. Coboundary (commutative case)
For an innitesimal k +1-simplex = (x0 . . . x +1) in the manifold M , its i'th face @ ()
(i = 0 . . . k + 1) is the innitesimal k-simplex obtained by deleting the vertex x . If now
! is a k-form with values in the vector space V , we dene its coboundary d! to be the
k + 1 form given by the usual simplicial formula
k
i
i
d!() =
X(;1) !(@ ()):
k +1
i
i=0
(4)
i
The fact that this expression vanishes when two vertices in are equal is clear: all but
two of the terms in the sum contain two equal vertices, so vanish since ! is a di erential
form, and the remaining two terms cancel out because of the sign (;1) and because ! is
alternating, by Proposition 3.1.
Clearly, d is a linear map (M V ) ! +1 (M V ).
The following fact is the standard calculation from simplicial theory, so the proof will
not be given.
i
k
k
Theory and Applications of Categories,
Vol. 2, No. 7
75
4.1. Proposition. For any k-form !, d(d(!)) = 0, the zero k + 2 form.
So we have the deRham complex (M V ) for M (with coecients in the vector space
V ). But formally, it is more like an Alexander-Spanier cochain complex from topology.
5. Wedge products
Let ! be a k-form on a manifold M with values in (the additive group of) a vector space
U , and let be an l-form on M with values in a vector space V . If now U V ! W is
a bilinear map into a third vector space W , we can manifacture a k + l-form ! ^ on M
with values in W as follows. We put
(! ^ )(x0 . . . xk+l ) := !(x0 . . . xk) (x0 xk+1 . . . xk+l ):
(5)
To see that it is a form, we have to see that the value is 0 if two of the x's are equal. Since
the question is local, it suces to consider the case where M = Rm and x0 = 0. When ! is
considered as a function of the remaining (x1 . . . xk ) 2 k D(m), it extends by Theorem
1.1 to a multilinear map ! : (Rm)k ! V , and similarly extends to a multilinear , and
then since is bilinear, the whole expression in (5) is multilinear in the the arguments
x1 . . . xk+l, thus vanishes if two of them are equal (the xi's being 1-neighbours of 0).
Remark. The reason we have chosen to let the -factor in (5) have an x0 in its rst
position, rather than the aesthetically more pleasing xk , is that then the formula also
makes sense for forms with values in vector bundles U , V and W over M , with now
being a brewise bilinear map. Then both factors in (5) are in the same bre, namely the
one over x0. It can be proved that for forms with values in constant bundles, we get the
same value in either case. For ease of future references, we display also the aesthetically
pleasing denition (equivalent to the above one, for constant bundles)
(! ^ )(x0 . . . xk+l ) := !(x0 . . . xk ) (xk xk+1 . . . xk+l):
(6)
It should also be remarked that our wedge product agrees with the \small" classical
one: there are two conventions on how many k!, l! and (k + l)! to apply. The \big" one
gives the determinant as dx1 ^ . . . ^ dxp, the small one gives p1! times the determinant as
dx1 ^ . . . ^ dxp, in other words, the volume of a simplex, rather than the volume of the
parallellepiped it spans. With the emphasis on simplices in our treatment, this is anyway
quite natural.
Since our notions (form, coboundary, wedge) agree with the classical ones in contexts
where the comparison can be made (models of SDG), the calculus of di erential forms
(d being an antiderivation w.r.to ^ etc.) holds, so there is no need to prove them in the
context here, except for simplication this task I therefore postpone - I believe it will be
calculations that one could copy from those of Alexander-Spanier cohomology. (Also, our
notions agree with the \linearized" ones employed in other treatises of forms in SDG, say
Koc81], MR91], Min88], or Lav96] this is essentially argued in Koc81], Section I.18,,
except for the wedge.)
Theory and Applications of Categories,
Vol. 2, No. 7
76
6. Fibre bundles
Let be a groupoid over (i.e. with for its space of objects). To make a space over ,
! , into a bre bundle for , means to0 provide it with a left action by . Similarly
it makes sense to talk about a map ! being a bre bundle homomorphism w.r. to
bre bundle structures for . If ! is a group bundle, there is an evident notion
that the -action consists of group homomorphisms, so is a \bre bundle of groups", or
a \group bundle" for . Similarly for bundles of vector spaces, etc., so that we have a
notion of \bre bundle of vector spaces", or just vector bundle, for .
To the groupoid is associated the group bundle consisting of its vertex groups ( )
for 2 it is actually a bre bundle of groups for , because arrows of acts from
the left on the endo-arrows of by conjugation: if 2 ( ) and : ! is an arrow
in , then f :=
;1 2 ( ). This group bundle for is what in the literature
Mac89] is called the gauge group bundle of we denote it
( ).
If ! ! is a homomorphism of groupoids over , any bre bundle for ! canonically
can be viewed as a bre bundle for also (in analogy with \restriction of scalars" for
modules over rings). In particular, the gauge group bundle of ! is canonically a group
bre bundle for .
M
E
M
M
M
E
E
E
M
x x
x
M
f
f
x x
f
x
y
y y
gauge
M
7. Dierential forms with bundle values
Let be a manifold, and : ! a bundle of Lie groups over . A dierential
-form on with values in : ! is a law which to any innitesimal simplex
( 0 . . . k ) in associates an element ( 0 . . . k ) in x0 (the bre over 0), in such
a way that if the simplex is degenerate (two vertices equal), then the value of is , the
neutral element in x0 .
We have that -forms are \alternating" in the restricted sense that swapping two of
the vertices i and j for 6= and 1 in the simplex implies a \sign change":
( 0 . . . i . . . j . . . k ) = ( 0 . . . j . . . i . . . k );1
(7)
The result of swapping 0 with i for 1 cannot be compared to the original value,
since these lie in di erent bres, in general. If the group bundle is \constant", i.e. of form
! for some Lie group , forms are also alternating in the sense of swapping
0 with i implies a sign change. This case is of course the same as the case of di erential
forms with values in the group . The proof of the \restricted" case is identical to the one
given above for Proposition 3.1. The sense in which is also changes sign when swapping
0 and i is dealt with in Section 14.
There is an evident way to multiply together two -forms with values in a group
bundle:
( )( 0 . . . k ) := ( 0 . . . k ) ( 0 . . . k )
The set of -forms in fact becomes a group k ( ) under this multiplication the neutral
element is the \zero form" given by ( 0 . . . k ) = x0 .
M
p
k
M
x
x
E
p
M
E
M
M
M
!
! x
k
x
E
x
!
E
k
x
x
! x
i
j
x
i j
x
x
G
x
M
x
x
M
! x
x
x
x
i
G
x
G
!
x
x
k
!
x
x
! x
k
x
x
E
z
z x
x
e
x
:
e
Theory and Applications of Categories,
Vol. 2, No. 7
77
Let be a Lie group. For -valued 1-forms , there is an evident notion of its
coboundary , which is a -valued 2-form:
( 0 1 2) := ( 0 1) ( 1 2) ( 2 0)
(8)
which we may think of as the \curve integral of around the boundary of the innitesimal
simplex ( 0 1 2)". (Warning: it di ers, for non-commutative , from the coboundary
of the Lie-algebra-valued 1-form to which it gives rise, by a correction term ^ we
return to this.)
Using that ( 2 0) = ( 0 2);1, and that an even permutation on the three arguments of does not change the value (since is alternating), we may rewrite the
denition (8) into
( 0 1 2) = ( 1 2) ( 0 2);1 ( 0 1)
(9)
For -valued -forms ( 2), it is also possible, but less evident, to dene its
cobundary +1-form it is less evident in the sense that there is no geometrically compelling way for the order in which to multiply together those factors ( 0 . . . ^ . . . +1),
whose product should constitute ( . . . +1), but it turns out (with proof in the spirit of Section 2) that these factors actually commute (for 2 !), so that the question of
order is irrelevant. We shall consider such \non-commutative" coboundary in Section 11.
For forms with values in a general group bundle ! , one needs one further
piece of structure in order to dene , namely a connection in ! . This is the topic
of the next section.
G
G
d!
!
G
d! x
x
x
! x
x
! x
x
! x
x
!
x
x
x
G
!
! x
x
! x
!
x
d!
d!
d! x
G
k
!
k
x
x
! x
x
! x
x
! x
x
:
k
d!
! x
d! xo
xi
xk
xk
k
!
E
d!
M
E
M
8. Connections
There!are two ways to dene connections, both of which are in terms of the re#exive graph
, the rst neighbourhood of the diagonal one is as an action of this graph on
(1) !
a bundle ! , the other is as a morphism of (re#exive) graphs (1) ! , where is
a groupoid with for its object manifold. The former can be reduced to the latter, by
taking to be the groupoid of isomorphism from one bre of ! to another one.
Here, we temporarily put the emphasis on the latter, groupoid theoretic, viewpoint (due
to Ehresmann).
So, a connection in a groupoid over is a homorphism of re#exive graphs r :
(1) ! , where
(1) denotes the rst neighbourhood of the diagonal. Equivalently, r
is a law which to any innitesimal 1-simplex ( ) associates an arrow r : ! in ,
in such a way that r is the identity arrow at . We shall assume throughout that
is a Lie groupoid, in the well known sense, cf Mac87]. The additive group bundle of a
vector bundle ! , as well as the groupoid GL( ) of a vector bundle considered in
Section 10, are Lie groupoids. The gauge group bundle of a Lie groupoid is a bundle of
Lie groups. Just as for forms, we may then prove that
r = r;1
(10)
One may also write r( ) instead of r .
M
M
E
M
M
M
E
M
M
M
M
x y
x
xx
E
yx
M
E
xy
y x
yx
yx
:
x
y
Theory and Applications of Categories,
Vol. 2, No. 7
78
8.1. Proposition. The set of connections in carries a canonical left action by the
group = 1(
) of 1-forms with values in the gauge group bundle of , and with
this action, it is a translation space over .
(The sense of the term 'translation space' will be apparent from the proof.)
Proof. Given a connection r in , and a 1-form with values in the gauge group
bundle, we get the connection r by putting
( r)( ) := ( )
r( )
(11)
(Note that ( ) is an endo-arrow at , so that the composition here does make sense.)
Given two connections r and ; in , their \di erence" ;
r;1 is the gauge-group-bundle
valued 1-form given by
(;
r;1)( ) := ;( )
r( ) = ;( )
(r( ));1
(12)
Then clearly
(;
r;1) r = ;
and it is unique with this property. The verications are trivial.
Remark. For drawings, the viewpoint of a connection r as an action by the graph (1)
on a bundle ! is usually more appropriate. If r and ; are two such, we present
here a drawing, which exhibits the action (via r and ;) of
on suitable points in ,
as well as the action of the \di erence" 1-form , (which now takes values in the group
bundle of automorphism groups of the bres):
C
G
gauge
G
!
!
!
x y
! x y
! x y
x y :
x
x y
x y
y x
x y
x y
:
M
E
M
x
y
E
!
HHrHyx
HH
( )
Hj
)?;xy
! x y
x
y
9. Combinatorial Bianchi identity
Given a connection r in a Lie groupoid , we dene its curvature as a 2-form with
values in the gauge group bundle of :
(
) = r( )
r( )
r( )
(13)
for (
) an innitesimal 2-simplex. Note that this is going around the boundary of
the innitesimal simplex (
) (in reverse order). We shall adopt a notational shortcut
in the proof of the following formula, by writing for r( ) : ! . Also, we omit
commas upper left indices denote conjugation. Then we have the combinatorial Bianchi
identity (essentially a form of the Homotopy Addition Lemma, cf. the Introduction):
R
R x y z
x y
y z
z x :
yx
y x
x y z
x y z
x
y
Theory and Applications of Categories,
Vol. 2, No. 7
79
9.1. Theorem. Let r be a connection in a groupoid , and let be its curvature. Then
for any innitesimal 3-simplex (
),
xy
)
( )
( )
( )
x= (
We shall below interpret the expression here as the coboundary or covariant derivative
of in a \complex" of group-bundle valued forms, with respect to r. This is still a
purely \combinatorial" gadget, but we shall later specialize to the case of the general
linear groupoid of a vector bundle with connection, and see that our formulation contains
the classical one.
R
x y z u
id
R yzu
R xyu
R xuz
R xzy :
R
Proof. With the streamlined notation mentioned, the identity to be proved is
xy
(
)
(
)
(
)
(
) = x
yz
zu
uy
yx
xy
yu
ux
xu
uz
zx
xz
zy
yx
id
the proof is now simply repeated cancellation: rst remove all parentheses, then keep
cancelling anything that occurs, or is created, which is of the form
etc., using (10)
one ends up with nothing, i.e. the identity arrow at .
The reader may like to see the geometry of this proof by drawing a tetrahedron with
vertices named
and then trace that path (of length 14 units) which the left hand
side of the above identity denotes cf. the description in the Introduction.
yx
xy
x
x y z u
10. Linear connections
We shall consider in particular linear connections in vector bundles ! they are
connections in the above groupoid theoretic sense, if we take the groupoid to be the
groupoid GL( ) of linear isomorphisms between the bres of ! . So for
in
, rxy : y ! x is a linear isomorphism, and rxx is the identity map of x. We then
have the following comparison, which we shall use later. Note that the rst term of the
left hand side is just the curvature (
) 2 ( x), applied to ( ). Since this is
2 ( x ), the left hand side itself is (log )(
), applied to ( ).
10.1. Proposition. Let be a section of the vector bundle ! , and let r be a
linear connection on it. Then for any innitesimal 2-simplex
rxy ryz rzx ( ) ; ( ) = rxy ryz ( ) ; rxz ( )
(14)
E
E
M
E
E
M
E
x
y
E
R x y z
e
M
GL E
GL E
R
f x
x y z
f x
f
E
M
x y z
f x
f x
f z
f z :
Proof. Since the question is local, we may assume that the bundle is V M
! for a
vector space , and we may assume that = Rm . Then there is a 1-form on with
values in ( ) such that
rxy ( ) = ( ( )( ) )
for all
2 and 2 . Now given a 2-simplex
and a section , there is a
unique linear map : = Rm ! such that ( ) = ( ) + ( ; ) for all 2 M1( ).
V
M
M
M
GL V
v y
x
y
M
v
g
M
x y
V
v
x
x y z
V
f u
f x
f
g u
x
u
x
Theory and Applications of Categories,
Vol. 2, No. 7
80
Rewriting r in terms of , the left hand side of (14) is simply (
)( ( )) ; ( ).
Writing ( ) = ( ) + ( ; ) and using linearity of ( ) : ! for any
, we
calulate the right hand side to be
( ) ( ) ( ); ( ) ( )+ ( ) ( ) ( ; ); ( ) ( ; )
(15)
Now since the values of are in ( )
( ) which is a vector space, there is
m
( ) such that ( ) = V + x( ; ). In particular,
a linear map x : R !
( ) ( ; ) = ( ; ) + x( ; ) ( ; ), and the last term here vanishes, because
it comes from a bilinear expression with ;
0 substituted in two places. So the last
term in our expression (15) is just ; ( ; ). One may similarly see that the second but
last term is ( ; ) (rewrite ( ; ) as ( ; )+ ( ; )). So the two last terms cancel.
Therefore (14) will follow if we prove in the ring
( ) that
( ) ( ) ( ); = ( ) ( ); ( )
(16)
Both sides clearly give 0 if the 2-simplex is degenerate, so both dene 2-forms with values
in (the additive group of)
( ). In particular, let denote the right hand side, so
(
)= ( ) ( ); ( )
.
Also, there is an
( ) valued 1-form on such that ( ) = + ( ), for
Then ^ is an
( )-valued 3-form, and therefore
0 = ( ^ )(
)= ( ) (
)
and hence
( )(
)= (
)
So we may multiply the expression dening (
) on the left by ( ) without changing the value. So the right hand side of (16) equals
( ) ( ) ( );
(17)
But the expression ( ) ( ) ( ) is a 2-form (
), hence alternating, so that
we may perform a cyclic permutation of the arguments
. Applying it to (17) gives
the left hand side of (16), proving the Proposition.
More generally, we may replace the section in the above Proposition by an -valued
-form, for any :
10.2. Proposition. Let ! be a vector bundle equipped with a connection r.
Let be an -form on with values in . Then for any innitesimal + 2-simplex
1 . . . n , we have
rxy ryz rzx ( 1 . . . n) ; ( 1 . . . n)
= rxy ryz ( 1 . . . n) ; rxz ( 1 . . . n)
f z
f x
d x y z
g z
x y y z f x
x
u v
x z f x
x z g z
x y y z g z
GL V
g z
x
x u
z
x g z
g z
x
g z
x
u
x z g z
v
x :
id
u
x
x
z
g z
x
V
f x
E nd E
E nd V
x
V
f x
x
x
g z
y
g y
x
E nd V
x y y z z x
id
x y y z
E nd V
x y z
x y y z
E nd V
x z :
x z :
M
u v
id
u v
u
E nd V
x z y z
x z x y z
z x x y z
x y z :
x y z
z x x y y z
z x x y y z
z x
id:
d x y z
x y z
f
n
E
n
E
!
x y z z
n
M
M
E
n
z
! x z
! z z
z
z
! x z
! z z
z
z
:
v
Theory and Applications of Categories,
Vol. 2, No. 7
81
Proof. The proof is the same as that of the previous Proposition, except that the linear
function g appearing there, and which is the degree 1 term of the Taylor series for the
map f : Rm ! V , is replaced by a linear map Rm ! Altn(Rm V ), again the degree 1
term of the Taylor series at x for !, viewed as a map from M = Rm to the vector space
Altn(Rm V ) of n-linear alternating maps from Rm to V .
11. Covariant derivative
We consider here a group bundle E ! M (typically the additive group bundle of a vector
bundle). Also, let there be given a connection r on it, i.e. a connection in the groupoid
GL(E ), or equivalently, an action by the neigbourhood graph M(1) !
! Mk on E ! M ,
acting by group homomorphisms. As in Section 7, we consider the group (E ) of k-forms
with values in E . We shall dene a map (not a group homomorphism, unless the group
bundle is commutative)
dr : k (E ) ! k+1 (E ):
Let ! 2 k (E ), and let (x0 . . . xk+1) be an innitesimal k + 1-simplex in M . As
usual, @i of such a simplex denotes the one obtained by omitting the i'th vertex. Then
we put
(dr !)(x0 . . . xk+1) = r(x0 x1)!(x1 . . . xk+1)
Y !(@i(x
k+1
1
0
. . . xk+1)1
where the sign in the exponent is ; if i is odd, + if i is even. (It turns out that for k 2,
the factors commute (essentially by the arguments at the end of Section 2), so that their
order is irrelevant.) For k = 0, and f 2 0(E ) (so f is a section of E ), dr (f ) is the
covariant derivative of f with respect to r,
dr (f )(x y) = (r(xy)f (y)) f (x);1:
(18)
If we consider a connection r in a Lie groupoid !
! M , we get a connection in
the group bundle gauge( ), since acts on gauge( ) by conjugation. We denote this
connection adr.
We now have the following reformulation of the combinatorial Bianchi identity (Theorem 9.1):
11.1. Theorem. Let r be a connection in a Lie groupoid !
! M , and let R be its
2
ad
r
curvature, R 2 (gauge( )). Then d (R) is the \zero" 3-form, i.e. takes only the
neutral group elements in the bres as values.
Proof. Let x y z u form an innitesimal 3-simplex. We have by denition of dadr that
(dadr(R))(xyzu) = adrxy R(yzu)
R(xzu);1
R(xyu)
R(xyz);1
Theory and Applications of Categories,
Vol. 2, No. 7
82
(omitting commas for ease of reading). Now the two middle terms may be interchanged,
by arguments as those of Section 2. We then get the expression in the combinatorial
Bianchi identity in Theorem 9.1, and by the Theorem, it has value x.
It is not true that r
r is the \zero" map, unless the curvature of the connection
vanishes see Section 13 below. To make the comparison with classical curvature, we
calculate here r
r : 0( ) ! 2( ) for the case of a commutative group bundle
! (which we write additively):
11.2. Proposition. Let 2 0( ), so is a section of the bundle , and let (
)
be an innitesimal 2-simplex. Then
id
d
d
E
d
d
E
E
M
f
E
r (dr (f )) (x
d
f
y z
E
x y z
) = rxy ryz ( ) ; rxz ( )
f z
f z :
For an -valued 1-form, r (
) = ( ) ; ( ) + rxy ( ). Now
let be given by the expression (18), but written additively, so ( ) = r ( ) =
rxy ( ) ; ( ) then we get, using additivity of rxy , six terms, four of which cancel, and
the two remaining ones give the expression claimed.
We recognize the right hand side here as the one we have in Proposition 10.1. So we
get by combining Propositions 10.1 and 11.2:
11.3. Proposition. For a section of the vector bundle with a linear connection r,
we have
(log )(
)( ( )) = ( r r )(
)
where is the combinatorial curvature of r, 2 2 ( ( )).
The \classical" curvature of a connection r in a vector bundle is usually dened in
terms of the right hand side of this equation. So the Proposition establishes the comparison that the logarithm of the combinatorial curvature equals the classical curvature.
Proof.
!
E
d
! x y z
! x y
! x z
!
! y z
! x y
f y
d
f
x y
f x
f
E
R
x y z
f x
d
R
d
f
R
x y z
GL E
12. Classical Bianchi Identity
We consider a vector bundle ! equipped with a linear connection r. Then we
get induced connections r in the group-, resp. ring-bundle ( ),
( ). We shall
consider diagrams, for 1,
E
M
ad
GL E
k
(
k
( ))
GL E
adr
d
>
k+1(
k (log)
(
k
_
( ))
GL E
k+1 (log)
( ))
E nd E
>
adr
d
(
k+1
_
( ))
E nd E
:
E nd E
Theory and Applications of Categories,
Vol. 2, No. 7
83
Even though, for 2 , log : ( x ) !
( x) is only partially dened (namely
on the respective monads M1( x), at least), the vertical maps are globally dened, since
for any form 2 k ( ( )), its values are
x . We shall address the question of
commutativity of these diagrams. The (partial) map log is equivariant with respect to the
connections r in ( ) and
( ), but it is not a group homomorphism ( x) !
( x), not even on M1( x) due to this, there is no \cheap" reason why these diagrams
should commute. However, log has the homomorphism property with respect to pairs of
commuting elements in ( x ) since the factors that dene adr for ( ) do commute
for 2, as stated also in Section 11, we conclude that the square above actually does
commute for 2.
Using the commutativity for = 2, and the combinatorial Bianchi identity (in the
form of Theorem 11.1), we shall prove
12.1. Theorem. (Classical Bianchi Identity) Let denote the curvature of a linear connection r in a vector bundle ! . As an element 2 2 ( ( )), adr ( ) is 0.
Proof. Consider the combinatorial curvature 2 2 ( ( )) of the connection r. By
the combinatorial Bianchi Identity, in the guise of Theorem 11.1, it goes to the \zero"
form by adr, hence also to the zero form by 3(log)
adr. Chasing the other way
round in the square rst gives log , which by Proposition 11.3 is the classical curvature
of r, so from the commutativity of the square follows adr( ) = 0.
For = 1, the square does not commute, in general.This fact is related to the MaurerCartan formula. For the case of a constant Lie group bundle, this relationship was considered synthetically in Koc82]. Here we do the more general case of a group bundle but, on the other hand, it is more special in another direction, since it deals only with a
group bundle of form ( ).
12.2. Proposition. Let 2 1( ( )), where ! is a vector bundle with a
connection r. Then
log( adr ( )) = adr(log ) + log ^ log
x
M
GL E
E nd E
e
!
ad
GL E
e
GL E
E nd E
E nd E
GL E
e
GL E
d
!
GL E
k
k
k
R
E
R
M
R
E nd E
d
R
GL E
d
d
R
R
R
d
R
k
GL E
!
GL E
d
!
E
d
!
M
!
!:
(The wedge here is with respect to the multiplication
in the ring
( ), which is
non-commutative so there is no reason for the wedge of a 1-form with itself to vanish.)
Proof. Let us write ( ) = x + ( ), so = log 2 1 (
( )). Let us write
conjugation by r( ) by an upper left index . Then for an innitesimal 2-simplex
, we have
r (
) =xy ( y + ( ))
( x ; ( ))
( x + ( ))
E nd E
! x y
e
x y
x y
!
E nd E
xy
x y z
d
! x y z
Multiplying out, we get
e
y z
e
x z
e
x y
+xy ( ) ; ( ) + ( )+
)
( ) +xy ( )
( ) ; ( )
(
ex
;xy (
yz
x z
yz
x z
y z
x y
x y
x z
x y
)
:
Theory and Applications of Categories,
Vol. 2, No. 7
84
plus a threefold product, which is easily seen to vanish (e.g. by coordinate calculations
like the following). The rst line in the formula is ( x + r )(
). We prove that
the second line equals ( )
( ) = ( ^ )(
). We claim that each of the three
terms in the second line give plus or minus ( ^ )(
) (twice plus, once minus). Let
us consider the rst only, the two others are similar. Since the question is local, we may
assume that = for a vector space, and that = Rm and we may assume
that ( ) = ( ; ) for
, with : !
( ) linear in the second
variable. Let = + 1 and = + 2. Then
e
x y
x z
E
u v
M
V
f u v
y
v
z
xy (y
z
x y z
x y z
M
u
d
x y z
V
u
x
d
f
x
)
(
x z
M
M
E nd V
d1 d2
; 1)
(
d
) =xy ( +
f x
f x d2
d
)
expanding the rst term out by linearity in the second variable of , we get two terms,
one of which vanishes because of \bilinear occurrence" of 2, and we are left with xy ( +
in its rst variable, and get xy ( ; 1)
1 ; 1)
(
2 ). Now we Taylor expand
( 2) plus a term which vanishes because of \bilinear occurrence" of 1. Also, since we
are conjugating with r( ) 2 ( ), whose logarithm depends linearily on ; = 1,
( ; 1) is xed under this conjugation. We are left with
f
d
d
d
f x d
f x
f
f x
f x d
x y
f x
d
d
GL V
y
x
d
d
( ; 1)
(
f x
d
f x d2
)=; (
f x d1
)
(
f x d2
)=; (
x y
)
(
x z
) = ;( ^ )(
)
x y z :
All said, we conclude that
adr ! (x
d
y z
)= x+ (
e
d x y z
) + ( ^ )(
x y z
)
where = log( ). Now subtracting x from the right hand side of this expression gives
something which is 0 2
( x) (in coordinates, it depends linearily on ; , say)
and this something is therefore the logarithm. This proves the Proposition.
!
e
E nd E
13. Analyzing dr
y
x
dr
We sketch in this section how the curvature enters in describing how the composite r
r
fails to be the zero map in the complex of -valued forms (where ! is a vector
bundle equipped with a linear connection r). This is classical, and of importance in
constructing the characteristic classes for the bundle ! .
Since the ring bundle ev ( ) ! acts in a bilinear way on ! simply by
evaluation
( ) M ! , the wedge product ^ is a well dened -valued
( )-valued 2-form, say
+ 2 form whenever is an -valued -form, and an
the (classical) curvature of the connection r. This is what denotes in the following
Proposition.
13.1. Proposition. Let be an -valued -form in the vector bundle . Then
d
E
E
E
E nd E
E nd E
k
E
M
M
E
R
E
!
E
M
R
k
M
!
E
E nd E
R
!
E
k
r (dr (! )) = R ^ !:
d
E
d
Theory and Applications of Categories,
Vol. 2, No. 7
85
We shall do the case = 1 only. The calculation is much similar to the
one in Proposition 11.2 (for any 1, in fact). For = 1, we get 12 terms in
r ( r ( ))(
), where (
) is an innitesimal 3-simplex. These terms are
Proof.
k
k
d
d
!
x y z u
k
x y z u
rxy(ryz ( ) ; ( ) + ( ))
;(rxz ( ) ; ( ) + ( ))
+rxy ( ) ; ( ) + ( )
;(rxy ( ) ; ( ) + ( ))
(in the rst line, use linearity of rxy to get three terms). Ten of these twelve terms cancel
! z u
in pairs, and we are left with
! y u
! y z
! z u
! x u
! x z
! y u
! x u
! x y
! y z
! x z
! x y
rxy(ryz
) ; rxz (
(
! z u
This equals, by Proposition 10.2, (with
rxyryz rzx
z1
)
! z u :
= )
(
u
); (
! x u
)
! x u :
The rst term here is the combinatorial curvature (
) of r, applied to ( ), so
)
when we subtract ( ), we obtain its logarithm, i.e. the classical curvature (
(applied to ( )). But by the denition (5) of wedge, (with evaluation
( ) M !
as the bilinear map),
R x y z
! x u
R x y z
! x u
! x u
E nd E
E
E
(
R x y z
)( (
)) = (
! x u
R
^
!
)(
)
x y z u :
14. Curvature dierence
We consider in this section a Lie groupoid !
! . When we have two connections;r1
and ; in , we may form their \di erence", which is a
( )-valued 1-form ;
r ,
cf. Section 8. So (;
r;1)( ) = ;xy
ryx. Also, we have two
( )-valued 2-forms,
r
;
namely their curvatures and .
14.1. Proposition. We have
adr (;
r;1) = ( r );1
;
(19)
M
gauge
x y
R
gauge
R
d
R
R :
We rst prove a Lemma which in some sense expresses how certain bundle valued
forms also are alternating with respect to interchanging the rst vertex with another one:
Proof.
Theory and Applications of Categories,
Vol. 2, No. 7
86
14.2. Lemma. Let be a 2-form with values in the gauge group bundle of a Lie groupoid
. For any connection r in , we have
r( ) (
)= (
)
The similar result holds for any -form with 1, using cyclic permutation of the
vertices, and inserting the exponent (;1) .
z x
x y z
k
z x y :
k
k
Proof. Since a Lie groupoid is locally trivial (cf. Mac87]) and the question is local, we
may assume that =
2-form ,
M
G
M
for a Lie group , and that is given by a -valued
G
G
(
)=( (
) )2
and also that r is given by a -valued 1-form
r =( ( ) )
Then the question reduces to whether (
) (= (
), since is alternating by
Proposition 3.1) is xed by conjugation by ( ), i.e. whether (
) and ( )
commute. This is so, by arguments similar to those of Section 2 (bilinearity in 2 =
; ).
To prove (19), let
form an innitesimal 2-simplex. We want to prove (omitting
commas for ease of reading)
r (;
r;1 )( ) = ( r ( ));1
; ( )
We rewrite the left hand side, by using the Lemma twice (rst for the connection ;, then
for the connection r) and get
;( )r( )
r (;
r;1 )( )
where upper left index denotes \conjugation by". Now, expanding the left hand side into
its constituents, using the denition of covariant derivative r , we get
; r ((r (; r )r )(r ; )(; r ))r ;
where the parentheses are just meant as an aid for the checking of the correctness of the
expansion. Removing all parentheses, and cancelling anything of the form r r (and
similarly for ;) that occurs or is created, we end up with
r r r ; ; ; = r ( ) ;( ) = ( r ( ));1 ; ( )
This proves the Proposition.
We shall use this Proposition in order to prove that the trace of r (and its wedge
powers) only depends on r up to a coboundary, so its deRham cohomology class does
not depend on the choice of connection.
x y z
x x y z
G
x
M
G
M
!
x ! x y
xy
y :
x y z
z x y
! x z
x y z
! x z
d
z
x
z x y
ad
d
zxy
zy
yx
R
zxy
ad
d
R
zxy :
xyz
ad
d
zy
yx
xy
yz
zy
yx
xz
zx
xy
yx
xy
yz
yx
zy
yx
xz
zx
xy
yz
R
zyx R
zxy
R
zxy
R
R
zxy :
xy
Theory and Applications of Categories,
Vol. 2, No. 7
87
15. Characteristic classes
We sketch how the theory of connections in vector bundles leads to characteristic classes.
This is the classical Chern-Weil theory, and there is not much specically synthetic about
the way we present it here.
For a vector bundle ! , we have for each 2 the linear trace map
E
M
x
M
( )!R
Trace :
E nd Ex
and collectively, they dene a linear map of vector bundles
Since for any linear isomorphism : x ! y , and any
Trace (
;1), it follows that the vector bundle map
a
E
E
( ) ! R over .
2 ( x), Trace ( ) =
E nd E
a
( )!
E nd E
M
E nd E
Trace :
M
a
R
M
is equivariant for the groupoid = GL( ) (with the groupoid acting by conjugation on
( ), and trivially on R). In particular, if r is a linear connection in ! ,
we get homomorphisms
E
E nd E
M
E
( )) ! k (
Trace : k (
E nd E
M
R) = (
k
M
M
)
(20)
commuting with the 's (respectively adr and the usual deRham ).
From the Bianchi identity, in the guise of Theorem 11.1, we therefore get that the
trace of the (classical) curvature of r is a cocycle in 2( ).
Since Trace :
( x ) ! R does not preserve the multiplicative structure, the
induced maps (20) will not preserve wedge products, in general. However, let 2
1( ( )). Then
Trace ( ^ ) = 0
For the left hand side, applied to an innitesimal 2-simples
, yields
d
d
d
R
M
E nd E
!
E nd E
!
!
:
x y z
Trace (
!
^
!
)(
x y z
) = Trace ( (
! x y
)
(
! x z
)) = Trace ( (
! x z
(the last equality by the fundamental property of trace that Trace (
and then we continue the equation
= Trace ((
!
^
!
)(
x z y
)) = ;Trace ((
!
^
!
)(
a
)
(
! x y
))
) = Trace (
x y z
b
b
)),
a
))
since forms are alternating. But then the total equation implies that Trace ( ^ )(
)=
0.
15.1. Theorem. Let r and ; ber two linear
connections on the vector bundle !
, with (classical) curvatures
and ; , respectively. Then the deRham cocycles
Trace ( r ) and Trace ( ; ) 2 2( ) dene the same cohomology class.
!
!
x y z
E
M
R
R
R
M
R
Theory and Applications of Categories,
Vol. 2, No. 7
88
Let = ;
r;1 be the \di erence" 1-form of the two connections, viewed as
connections with values in the groupoid GL( ). Thus, for the combinatorial curvatures
r and ; , we have, by Proposition 14.1,
Proof.
!
E
R
R
adr
d
( ) = ( r );1
!
R
R
;
now apply log the values of r and ; commute, as observed in Section 12, so that log
has the homom